Weak Convergence to the Matrix Stochastic Integral BdB

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form ∫01WdW, where W(r) is standard Brownian motion. In multiple regressions and vector autoregressions with vector ARIMA processes the theory involves weak convergence to matrix stochastic integrals of the form ∫01BdB’, where B(r) is vector Brownian motion with non scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to ∫01BdB’ under quite general conditions. The theory is applied to vector autoregressions with integrated processes.

Keywords: Integrated process, invariance principle, Near integrated time series; Stochastic integral, Vector autoregression, Weak convergence